Does there exist a Noetherian domain (which is not a field) whose field of fractions is (isomorphic with) $\mathbb C$?

Question

Does there exist a Noetherian domain which is not a field whose field of fractions is (isomorphic with) $\mathbb C$ ?

Answer 1

If there is such a Noetherian domain $R$ of positive dimension, we may assume it to be one-dimensional by passing to a localization at a height one prime. Passing further to the integral closure of $R$ within ${\mathbb C}$ (which is again Noetherian by the Krull-Akizuki Theorem), we may even assume $R$ to be normal, hence a discrete valuation ring corresponding to a discrete valuation $\eta: {\mathbb C}^{\times}\to {\mathbb Z}$.

However, there is no discrete valuation on ${\mathbb C}$, because as explained in every field of characteristic 0 has a discrete valuation ring? the value group of each valuation of ${\mathbb C}^{\times}$ needs to be divisible due to the presence of arbitrarily high roots in ${\mathbb C}$.